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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-clel3gALT | Structured version Visualization version GIF version |
Description: Alternate proof of clel3g 3646. (Contributed by BJ, 1-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-clel3gALT | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2810 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐵) | |
2 | 1 | biantrurd 532 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵))) |
3 | 19.41v 1946 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵)) | |
4 | 2, 3 | bitr4di 289 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵))) |
5 | eleq2 2817 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) | |
6 | 5 | bicomd 222 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝑥)) |
7 | 6 | pm5.32i 574 | . . 3 ⊢ ((𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ (𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)) |
8 | 7 | exbii 1843 | . 2 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)) |
9 | 4, 8 | bitrdi 287 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 |
This theorem is referenced by: (None) |
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